Cosmology Cosmic Aging Amplitude Sharp Desi Sharp Threshold Band

A tighter test for cosmic aging

In cosmology, the expansion of the universe is often described by an equation of state parameter, w, which relates pressure to energy density. For a cosmological constant, w is exactly -1. The framework called Recognition Science (RS) proposes a small deviation, δw, from this value, driven by a quantity it calls cosmic Z, a time-varying scalar field. The challenge is to predict how large δw should be, and to state what observation would disprove that prediction.

The new theorem, desi_sharp_threshold_band, establishes a precise numerical window for this deviation. It proves that the sharpened threshold for δw lies between 0.026 and 0.034. This is a factor of three to four times tighter than the older, looser ceiling of about 0.118, which was derived from a different, more extreme assumption about the efficiency of a cosmic heat engine. The theorem is a formal result in the framework's machine-checked library of formal theorems, meaning the calculation is verified step by step.

The tighter band comes from a specific comparison. The framework's model predicts a certain value for the dark energy density parameter, ΩΛ, which differs from the central value measured by the Planck satellite in 2018. The gap between these two numbers, about 0.0044, is what the proposed deviation δw must explain. The new theorem translates this required deviation into a concrete, testable range, and then sets the falsifier threshold at twice the upper end of that range, giving a clean target for experiments.

This matters because it turns a vague prediction into a sharp one. Under the older ceiling, a null result from the Dark Energy Spectroscopic Instrument (DESI) at δw = 0.05 would not have ruled out the framework's idea. Under the new threshold, a null result at δw = 0.005 would directly contradict the framework's explanation of the Planck-RS gap. The theorem also proves that this new threshold is comfortably below the old ceiling, confirming it is a genuinely more powerful test.

What the theorem does not do is claim that the framework's model is correct, or that the deviation δw actually exists. It only states what the framework's own assumptions imply for the size of that deviation, and what experiment would falsify it. The translation from the observed ΩΛ gap to the implied δw relies on a linear approximation, which the framework itself flags as a hypothesis requiring further work. The theorem is a statement about the internal consistency of a model, not a measurement of the universe.

/-- The sharpened threshold is in `(0.026, 0.034)`. -/
theorem desi_sharp_threshold_band :
    0.026 < desi_sharp_threshold ∧ desi_sharp_threshold < 0.034 := by
  unfold desi_sharp_threshold
  have h1 : 0.013 < delta_w_implied_max := delta_w_implied_max_gt
  have h2 : delta_w_implied_max < 0.017 := delta_w_implied_max_lt
  refine ⟨?_, ?_⟩ <;> linarith

/-- **The sharpened threshold is at least a factor 3 tighter than
the Carnot ceiling.** -/
theorem desi_sharp_tighter_than_carnot :
    desi_sharp_threshold < Cost.Jcost phi / 3 := by
  -- desi_sharp_threshold < 0.034 from band
  have h := desi_sharp_threshold_band.2
  -- J(φ) > 0.11
  have h_jphi : 0.11 < Cost.Jcost phi := by
    rw [jcost_phi_closed]
    have hphi : 1.61 < phi := phi_gt_onePointSixOne
    linarith
  have : (0.034 : ℝ) < 0.11 / 3 := by norm_num
  linarith

/-- Numerical: the observed gap satisfies `0.004 < Δ_obs < 0.005`. -/
theorem gap_observed_band : 0.004 < gap_observed ∧ gap_observed < 0.005 := by
  unfold gap_observed planck_omega_lambda_central rs_omega_lambda_mid
  refine ⟨?_, ?_⟩ <;> norm_num

/-- The strong falsifier is below the smallest implied `δw_min`. -/
theorem strong_falsifier_below_implied_min :
    bit_strong_falsifier < delta_w_implied_min := by
  unfold bit_strong_falsifier
  exact lt_trans (by norm_num : (0.005 : ℝ) < 0.008) delta_w_implied_min_gt

/-- The implied `δw` at the midpoint. -/
def delta_w_implied_mid : ℝ := gap_observed / z_norm_late_mid

What this page does not claim

The theorem does not claim that the framework's model is correct or that the deviation δw exists in nature. The theorem does not claim that the linear approximation used to translate the ΩΛ gap is an exact solution of the framework's equations. The theorem does not claim that a null result at δw = 0.005 would falsify all possible explanations of the cosmic-aging amplitude, only the specific BIT explanation.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/CosmicAgingAmplitudeSharp.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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